# Unicursal curve

A plane curve $\Gamma$ which may be traversed such that the points of self-intersection are visited only twice. For a curve to be unicursal it is necessary and sufficient that there are at most two points through which there pass an odd number of paths. If $\Gamma$ is a plane algebraic curve of order $n$ having the maximum number $\delta$ of double points (including improper and imaginary ones), then $\delta = ( n - 1) ( n - 2)/2$( where a point of multiplicity $k$ is counted as $k ( k - 1)/2$ double points).

Every integral $\int R ( x, y) dx$, where $y$ is the function of $x$ defined by the equation $F ( x, y) = 0$ giving an algebraic unicursal curve and $R ( x, y)$ is a rational function, can be reduced to an integral of a rational function and can be expressed in terms of elementary functions.

In algebraic geometry, a unicursal curve $U$ is a rational curve, i.e. a curve that admits a parametric representation $x = \phi ( t)$, $y = \psi ( t)$ with $\phi$ and $\psi$ rational functions. Such a curve is an algebraic curve of effective genus $0$. For every irreducible curve $\Gamma$ there exists a birationally equivalent non-singular curve $\widetilde \Gamma$. This $\widetilde \Gamma$ is unique up to isomorphism. The genus of $\widetilde \Gamma$ is called the effective genus of $\Gamma$. The unicursal curves are the irreducible algebraic curves of effective genus zero. This (more or less) agrees with the general geometric definition above, in that the parametrization provides a "traversion" .